球面上纯 p-自旋模型在任意能量下复杂度的集中性
概率论
2024-06-19 v1
摘要
我们考虑球面纯 p p p H N ( σ ) = 1 N ( p − 1 ) / 2 ∑ i 1 , . . . , i p = 1 N J i 1 , . . . , i p σ i 1 ⋯ σ i p H_{N}\left(\boldsymbol{\sigma}\right)=\frac{1}{N^{\left(p-1\right)/2}}\sum_{i_{1},...,i_{p}=1}^{N}J_{i_{1},...,i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}} H N ( σ ) = N ( p − 1 ) /2 1 ∑ i 1 , ... , i p = 1 N J i 1 , ... , i p σ i 1 ⋯ σ i p σ = ( σ 1 , . . . , σ N ) ∈ S N − 1 : = { σ ∈ R N : ∥ σ ∥ 2 = N } \boldsymbol{\sigma}=\left(\sigma_{1},...,\sigma_{N}\right)\in \mathbb{S}^{N-1}:=\left\{ \boldsymbol{\sigma}\in\mathbb{R}^{N}:\,\left\Vert \boldsymbol{\sigma}\right\Vert _{2}=\sqrt{N}\right\} σ = ( σ 1 , ... , σ N ) ∈ S N − 1 := { σ ∈ R N : ∥ σ ∥ 2 = N } J i 1 , . . . , i p J_{i_{1},...,i_{p}} J i 1 , ... , i p p ≥ 32 p\geq 32 p ≥ 32 E > − E ∞ E>-E_\infty E > − E ∞ E ∞ E_\infty E ∞ H N ( σ ) ≤ N E H_N(\boldsymbol{\sigma})\leq NE H N ( σ ) ≤ N E σ \boldsymbol{\sigma} σ 1 1 1 E E E
引用
@article{arxiv.2109.03163,
title = {Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies},
author = {Eliran Subag and Ofer Zeitouni},
journal= {arXiv preprint arXiv:2109.03163},
year = {2024}
}